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A083846
a(n) is the largest prime of the form x^2 + 1 <= 10^n.
5
5, 37, 677, 8837, 98597, 972197, 9985601, 99800101, 999444997, 9999200017, 99986234437, 999920001601, 9999799764517, 99999200001601, 999999202999697, 9999993200001157, 99999979750774757, 999999848000005777
OFFSET
1,1
COMMENTS
It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence does not become a constant, but this has not been proved. It is easily shown that all terms greater than 5 end in 1 or 7.
REFERENCES
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.
LINKS
Eric Weisstein's World of Mathematics, Landau's Problems.
MATHEMATICA
Do[ k = Floor[ Sqrt[ 10^n] - 1]; While[ !PrimeQ[k^2 + 1], k-- ]; Print[k^2 + 1], {n, 1, 19}]
lpf[n_]:=Module[{p=NextPrime[10^n, -1]}, While[!IntegerQ[Sqrt[p-1]], p= NextPrime[ p, -1]]; p]; Array[lpf, 10] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Feb 11 2023 *)
KEYWORD
nonn
AUTHOR
Harry J. Smith, May 05 2003
EXTENSIONS
Edited and extended by Robert G. Wilson v, May 08 2003
STATUS
approved